One of the general rules of digital signal processing is that sampling of a typical analog signal should be performed above the Nyquist rate. The Nyquist rate is defined as twice the highest frequency (referred to as the Nyquist frequency) of the signal to be sampled. An equivalent statement is that for a given sampling rate, all signal content at and above half this sample rate should be removed before being sampled. If this rule is followed, then the analog signal can be uniquely determined or reconstructed from the samples taken. This relationship is known as the Nyquist sampling theorem. If this rule is not followed, a phenomenon known as aliasing occurs, in which tones at frequencies above half the sample rate appear at other frequencies in the spectrum of the sampled signal.
Aliasing is the direct result of the process of sampling an analog signal to produce a digital signal. Sampling theory shows that the spectrum of the resulting digital signal will be the sum of periodically repeated copies of the spectrum of the analog signal. Specifically, the analog spectrum is shifted by nF.sub.s, where n is an integer and F.sub.s is the sampling frequency, to produce an infinite number of shifted copies which are then summed together to produce the spectrum of the digitized signal. If the Nyquist theorem is obeyed, the spectrum of the digitized signal will be identical in the range from zero to F.sub.s /2 Hz to that of the analog signal. If the Nyquist theorem is not obeyed, the spectrum value at each frequency F.sub.o, in the range from zero to F.sub.s /2 Hz, will be equal to the sum of the spectrum value at each frequency nF.sub.s -F.sub.o and nF.sub.s +F.sub.o in the analog signal.
A tone detector may be designed to decide whether or not a tone at frequency F.sub.o is present. Aliasing may cause the detector to spuriously declare the presence of the F.sub.o tone both for a valid tone at F.sub.o and for any tones at nF.sub.s -F.sub.o or nF.sub.s +F.sub.o. FIG. 1 illustrates a frequency spectrum of a sampled signal which consists of an actual tone at frequency Fo, and its aliases at F.sub.s -F.sub.o and F.sub.s +F.sub.o, 2F.sub.s -F.sub.o and 2F.sub.s +F.sub.o, etc.
A known method used to prevent aliasing is to place an analog low-pass filter (LPF), having a cutoff frequency less than F.sub.s /2, in front of a digitizer. Referred to as anti-aliasing filtering, this practice insures that the Nyquist theorem is obeyed, preventing aliasing problems. However, the necessity for this anti-aliasing filter with such a cutoff frequency may add considerable cost to a product, and it may occupy precious real estate on a printed circuit board (PCB). In other cases, where hardware is being reused for another application, the anti-aliasing filters may need to be changed or new filters may need to be added.